Let $k(n)$ be the maximal $k$ such that there exists $m\le n$ such that each of the integers$$m+1,\dots ,m+k$$are divisible by at least one prime $>k$. Estimate $k(n)$ - in particular, is it true that$$\log k(n)\le (\log n{)}^{1/2+o(1)}?$$

AMS 11 · open (literature)

theorem erdos_962 :
    answer(sorry) ↔
      ∃ ε : ℕ → ℝ,
        (∀ δ > 0, ∀ᶠ n in atTop, |ε n| < δ) ∧
        ∀ᶠ n : ℕ in atTop,
          log (k n : ℝ)
            ≤ rpow (log n) ((1 : ℝ) / 2 + ε n)